# Category:Intervals

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This category contains results about Intervals in the context of Order Theory.

Definitions specific to this category can be found in Definitions/Intervals.

Let $\struct {S, \preccurlyeq}$ be an ordered set.

Let $a, b \in S$.

The **intervals between $a$ and $b$** are defined as follows:

### Open Interval

The **open interval between $a$ and $b$** is the set:

- $\openint a b := a^\succ \cap b^\prec = \set {s \in S: \paren {a \prec s} \land \paren {s \prec b} }$

where:

- $a^\succ$ denotes the strict upper closure of $a$
- $b^\prec$ denotes the strict lower closure of $b$.

### Left Half-Open Interval

The **left half-open interval between $a$ and $b$** is the set:

- $\hointl a b := a^\succ \cap b^\preccurlyeq = \set {s \in S: \paren {a \prec s} \land \paren {s \preccurlyeq b} }$

where:

- $a^\succ$ denotes the strict upper closure of $a$
- $b^\preccurlyeq$ denotes the lower closure of $b$.

### Right Half-Open Interval

The **right half-open interval between $a$ and $b$** is the set:

- $\hointr a b := a^\succcurlyeq \cap b^\prec = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \prec b} }$

where:

- $a^\succcurlyeq$ denotes the upper closure of $a$
- $b^\prec$ denotes the strict lower closure of $b$.

### Closed Interval

The **closed interval between $a$ and $b$** is the set:

- $\closedint a b := a^\succcurlyeq \cap b^\preccurlyeq = \set {s \in S: \paren {a \preccurlyeq s} \land \paren {s \preccurlyeq b} }$

where:

- $a^\succcurlyeq$ denotes the upper closure of $a$
- $b^\preccurlyeq$ denotes the lower closure of $b$.

## Subcategories

This category has the following 2 subcategories, out of 2 total.

## Pages in category "Intervals"

The following 2 pages are in this category, out of 2 total.